Square each integer $n$ in the range $1\le n\le 10$ and find the remainders when the squares are divided by $11$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $11$?
Answer: First notice that $(11-n)^2=11^2-2\cdot 11+n^2\equiv n^2\pmod{11}$, and since we're asked to find distinct results, we only need to compute the squares of $n=1,2,3,4,5$. Respectively, $n^2\equiv 1,4,9,5,3\pmod{11}$. Thus, $1+4+9+5+3=22=11\cdot\boxed{2}$.